Closing the Window on Warm Dark Matter

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Steen H. Hansen1, Julien Lesgourgues2, Sergio Pastor3, Joseph Silk1 1 Department of Physics, Nuclear & Astrophysics Laboratory, University of Oxford, Keble Road, Oxford OX1 3RH, U.K. 2 Laboratoire de Physique Théorique LAPTH, B.P. 110, F-74941 Annecy-le-Vieux Cedex, France 3 Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, D-80805, Munich, Germany Sterile neutrinos may be one of the best Warm Dark Matter candidates we have today. Both lower and upper bounds on the mass of the sterile neutrino come from astronomical observations. We show that the proper inclusion of the neutrino momentum distribution and the solution of the kinetic equations with the correct coherence breaking terms lead to the near exclusion of the sterile neutrino as a dark matter candidate.

PACS number(s): 14.60.Pq, 14.60.St, 95.35.+d

I. INTRODUCTION do massless neutrinos. Therefore they do not share the entropy release from the successive particle annihilations. Astrophysics provides an increasing amount of inde- Since they were relativistic at decoupling, their distribu- pendent indications that the dark matter of the universe tion function in momentum space is subsequently that of is warm, so that the small-scale fluctuations are damped a massless fermion, but with a temperature, TW , which out by free streaming. This is most easily achieved by is given today by giving a keV mass to the DM particle, in which case the 1/3 preferred candidate is the sterile neutrino. Support for 2 94 eV TW0 = Tν0 ΩW h , (1) warm dark matter (WDM) comes from simulations of the mW number of satellite galaxies [1] and of disk galaxy forma- 1 1 tion without the need for stellar feedback [2], which both where Tν0 1.946 K, H0 = 100 hkm s− Mpc− ,ΩW is find that a DM particle mass of about 1 keV is optimal: a the present≈ energy density of WDM in units of the criti- significantly larger mass has little impact on galaxy for- cal density, and mW is the WDM mass. The needed en- mation, and a significantly smaller mass would lead to tropy release in these conventional models is much bigger the well known difficulties faced by hot dark matter. A than allowed in the standard model, and such WDM can- quantitative lower limit on the candidate WDM particle didates should therefore have decoupled before a larger mass is inferred from the existence of a massive black hole gauge group breaks down. at large redshift [3] and the requirement of sufficiently Now a very natural candidate for the WDM parti- early galaxy formation to account for reionization of the cle is a massive sterile neutrino mixed with an ordi- universe and the observed Ly-α forest properties [4], con- nary neutrino [7–11]. Since the mixing angle is tempera- straining the DM mass to be larger than 0.75 keV. A re- ture dependent [12] (and for small vacuum mixing angle, 2 7 cent discussion of x-ray emission from decays of sterile sin 2θ 10− ), a small amount of these heavy neutrinos, neutrinos [5] has imposed an upper limit of about 5 keV relative∼ to ordinary active neutrinos, can be produced at on the neutrino mass. Here we discuss a reinterpretation high temperatures. The distribution function of sterile of these bounds on neutrino mass, and demonstrate that neutrinos is, to a fair approximation, characterized by solution of the kinetic equations with the correct coher- the temperature of massless neutrinos, but smaller by a ence breaking terms lead to the virtual exclusion of the factor . For a specific choice of mW and ΩW today the sterile neutrino as a dark matter candidate. value ofX can be found from X 2 94 eV 2 =ΩW h 10− , (2) II. WDM PARTICLE DECOUPLING X mW ∼ therefore the two models produce the same contribution All of these studies [1–4] are based on the mass- to ρ of WDM particles today if dependent cut-off on small scales, produced by free- tot streaming. In the previously cited studies, a “conven- T 3 tional” WDM model was considered for the underlying = W0 . (3) X T particle physics (see refs. [2,6] for recent overviews of such ν0 particle models). In such WDM models, the particles de- However, WDM particles with mW have a different dis- couple in the early Universe at higher temperatures than tribution function in these two models, and their free-

1 streaming effect is not equivalent [8]. We will use con- IV. UPPER BOUNDS ventional WDM (cWDM) when referring to the first case, and sterile neutrino WDM (sWDM) for the second one. Sterile neutrinos in the keV mass range have a de- cay time that is of cosmological interest. Recently a very interesting paper appeared [5], where the signature III. LOWER BOUNDS from decaying sterile neutrinos in galaxies and clusters of galaxies was studied in detail ∗. A bound m<5keV The two neutrino models are easily included in a Boltz- was derived, using the relation between mass and mix- man code, in order to compute the present matter power ing angle obtained in [10]. It is now very interesting to spectrum P (k). Using the code cmbfast [13], we have note that as shown in ref. [11] the required mixing angle found analytical fits for the transfer functions, T (k), re- in reality should be about a factor two larger to gener- lating the power spectrum in the WDM to the CDM ate the same cosmological energy density. This factor scenario of 2 was obtained by considering the exact form of the

W coherence breaking terms in the evolution equations for 2 P (k) the neutrino density matrix. Including this effect would T (k)= for ΩW =ΩCDM , (4) P CDM(k) further strengthen the bound on m<5 keV by approx- imately this factor of 2. Thus we see that the excluded W where P is the power spectrum for cWDM, and a region from above [3,4] and from below [5,11] overlap, al- ν similar expression with P (k)forthesWDMmodel. most completely excluding the sterile neutrino as a dark These transfer functions, which essentially reflect the free matter candidate. streaming cut-off, have the form However, the exclusion of sterile neutrinos as a dark

5/ν matter candidate cannot be said to be complete yet, for T (k)= 1+(αk)2ν − , (5) the following reasons. First, the temperature of the ster- h i ile neutrinos is really slightly lower than the active neu- 1 trino temperature, since the sterile neutrinos are being where k is the wavenumber in units hMpc− , ν =1.12, and α depends on the cosmological parameters as produced while the muons are still present in the Uni- verse, T 130 MeV [16]. Furthermore, the factor of 3-4 ≈ Ω b h c m d found above depends on the specific values of ΩW and h, α = A W W . (6) and can therefore change slightly. Lastly, and probably 0.3 0.65 500 eV most importantly, the rather simplistic combination of Numerically we find for cWDM, A =1.07, b =0.11, the results of ref. [10] with the factor 2 of ref. [11] de- c =1.20 and d = 1.11 in good agreement with [6]. serves a more careful investigation. Such an analysis is For the sWDM one can− derive similar numbers by noting unfortunately not completely trivial, since it demands the numerical solution of the non-linear equations describing that the mass in the cWDM case differs by (TW0 /Tν0 ). This means that if we have a dependence Ωb hcmd for the density matrix, taking into account the exact form W W of the coherence breaking terms. It is therefore crucial the sWDM case, and a dependence Ωb0 hc0 md0 for the W W to first reach a better understanding of the neutrino pro- cWDM case, then one finds duction mechanism.

b c d b0 d0/3 c0 2d0/3 d0+d0/3 ΩW h mW =ΩW− h − mW , (7) which is solved by b0 = b + d/4, d0 =3d/4andc0 = c + d/2, in good agreement with what we found numerically, by explicitly changing the massive neutrino phase-space distribution function like in ref. [14]. To be very explicit, this means that for a given cut-off scale of the power spectrum one can find the corresponding mass of the elementary particle, and the mass will dif- fer in the two cases. E.g. if for cWDM one finds mW = 1 keV, then this corresponds in the sWDM case to 2 1/3 mW =(ΩW h 94/1000)− keV 3.8 keV, when using ≈ ∗One could imagine a slight change of strategy in the anal- h =0.7andΩW =0.4. In other words, if one believes that sterile neutrinos indeed constitute the dark matter, ysis of ref. [5], namely to consider regions with large concen- tration of dark matter, but with little baryonic matter. Such and they are produced as described in refs. [7–11], then “dark blobs” may have been observed by inverting the mat- the bounds obtained in refs. [3,4], 0 75 keV, should mW > . ter distribution in clusters of galaxies from weak lensing, and really be multiplied by a factor 3.4, and the lower bound comparing with the baryonic matter inferred from optical ob- on sterile neutrinos as dark matter is thus about 2.6keV. servations of the cluster [15], but the significance of such blobs is far from being established.

2 ACKNOWLEDGEMENT

SHH and SP are supported by Marie Curie Fellowships of the European Commission under contracts HPMFCT- 2000-00607 and HPMFCT-2000-00445. They acknowl- edge a visit to LAPTH, supported by CNRS, during which part of this work was carried out.

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